**1. Introduction**

Nanofluids were first explained by Choi [1] in 1995. Nanofluids are a composition of nanoparticles and a base fluid including oil, water, ethylene-glycol, kerosene, polymeric solutions, bio-fluids, lubricants, oil, etc. The material of the nanoparticles [2] involves chemically stable metals, carbon in multiple forms, oxide ceramics, metal oxides, metal carbides, etc. The magnitude of the nanoparticles is substantially smaller (approx. less than 100 nm). Nanofluids have multitudinous applications in engineering and industry [3,4], such as smart fluids, nuclear reactors, industrial cooling, geothermal power extract, and distant energy resources, nanofluid coolant, nanofluid detergents, cooling of microchips, brake and distant vehicular nanofluids, and nano-drug delivery. In the light of these applications, numerical researchers discussed the nanofluids in different geometrical configurations. For instance, Gourarzi et al. [5] scrutinized the impact of thermophoretic force and Brownian motion on hybrid nanofluid. They concluded with the excellent point that nanoparticle formation on cold walls is more essential due to thermophoresis migration. Ghalandari et al. [6] used CFD to model silver/water nanofluid flow towards a root canal. The effects

Arain, M.B.; Bhatti, M.M.;Zeeshan, A.; Alzahrani, F.S. Bioconvection Reiner-Rivlin Nanofluid Flow between Rotating Circular Plates with Induced Magnetic Effects, Activation Energy and Squeezing Phenomena. *Mathematics* **2021**, *9*, 2139. https:// doi.org/10.3390/math9172139

**Citation:**

Academic Editor: Mostafa Safdari Shadloo

Received: 29 July 2021 Accepted: 31 August 2021 Published: 2 September 2021

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of injection height, nanofluid concentration, and the rate of volumetric flow were explored and addressed. Sheikholeslami and Vajravelu [7] studied the control volume-based finite element approach to determine magnetite nanofluid flow into the same heat flux in the whole cavity. The impact of Rayleigh number, Hartmann number, and volume friction of nanofluid flow magnetite (an iron oxide) and heat transfer features were discussed. Sheikholeslami and Ganji [8] addressed hydrothermal nanofluid in the existence of magnetohydrodynamics by using DTM. They discussed the impact of squeezing number and nanofluid volume fraction on heat transfer and fluid flow. Biswal et al. [9] deliberated fluid flow in a semi-permeable channel with the influence of a transverse magnetic field. Zhang et al. [10] considered the outcome of thermal diffusivity and conductivity of numerous nanofluids utilizing the transient short-hot-wire technique. Fakour et al. [11] inquired the laminar nanofluid flow in the channel using the least square approach with porous walls. This study shows that by enhancing Hartman and Reynolds number, the velocity of the nanofluid flow in the channel declines and an extreme amount of temperature is enhanced. More, enhancing the Prandtl number along with the Eckert number also increases the temperature distribution. Zhu et al. [12] inquired the second-order slip and migration of nanoparticles from a magnetically influenced annulus. They applied a well-known HAM technique for solving the equations, and a h-curve was drawn to validate the exactness of the obtained solution. Ellahi et al. [13] revealed the impact of Poiseuille nanofluid flow with Stefan blowing and second-order slip. The accuracy of the analytical solution is obtained by the HAM and verified by h-curve and residual error norm for each case. They claim that the ratio of buoyancy forces in the existence of a magnetic field played a vital role in velocity distribution.

Magnetohydrodynamic (MHD) has grabbed different researchers' attention because of its multitudinous applications in the agricultural, physics, medicine, engineering, and petroleum industries, etc. For instance, applications of MHD involve bearing sand boundary layer control, MHD generators, rotating machines, viscometry, electronic storing components, turbomachines, lubrications, oceanographically processes, reactor chemical vapor deposition, and pumps. The magnetic field plays an essential role in controlling the boundary layer of momentum and heat transfer. The presence of magnetics is beneficial to control fluid movement. It is worthwhile to mention that the magnetic essential modified the outcomes of heat transfer in the flow by maneuvering the suspended nanoparticles and reorganized the fluid concentration. Khan et al. [14] studied the magnetohydrodynamic nanofluid flow between the pair of rotating plates. Zangooee et al. [15] analyzed the hydrothermal magnetized nanofluid flow between a pair of radiative rotating disks. From their studies, it is perceived that concentration decreases while increasing in Reynolds number, but on the other hand, the temperature is increasing for Reynolds number. By enhancing the value of the stretching parameter, the Reynolds number increases at the upper disc and decreases at the lower plates. Hatami et al. [16] analytically inquired the magnetized nanofluid flow in the porous medium. These results showed that the magnetic field opposes fluid flow in all directions. In addition, they claimed that the action of thermophoresis increases temperature and reduces the flow of heat from the disc. Nanoparticles shape effect on magnetized nanofluid flow over a rotating disc embedded in porous medium investigated by Rashid and Liang [17]. Abbas et al. [18] studied a fully developed flow of nanofluid with activation energy and MHD. The study's main findings demonstrate that flow field and entropy rate are highly affected by a magnetic field. The results indicate that both the flow and entropy rates of the magnetic field are significantly affected. Rashidi et al. [19] inquired steady MHD nanofluid flow with entropy generation and due to permeable rotating plates. Alsaedi et al. [20] inquired the flow of copper-water nanofluid with MHD and partial slip due to a rotating disc. They contemplated water as a base fluid and copper nanoparticles. They concluded with the remark that for greater values of a nanoparticle volume fraction, the magnitude of skin friction coefficient had been increased both for radial and azimuthal profiles. Asma et al. [21] numerically discussed the MHD nanofluid flow over a rotating disk under the impact of activation energy.

They observed that the concentration and temperature both show a growing tendency by increasing Hartman numbers. Aziz et al. [22] inquired the three-dimensional motion of viscous nanoparticles over rotating plates with slip effects. They showed that concentration profile and temperature distribution show enhancing behaviors for increasing values of Hartmann number. Hayat et al. [23] numerically inquired the nanofluid flow because of rotating disks with slip effects and magnetic field. These studies showed that more significant levels of the magnetic parameter indicate reduced velocity distribution behavior, whereas temperature and concentration distribution show opposite behavior. The hydromagnetic fluid flow of nanofluid due to stretchable/shrinkable disk with non-uniform heat generation/absorption is inquired by Naqvi et al. [24]. The graphical results of the studies showed that the higher values of the Prandtl number give an improved temperature, but when thermophoresis and Brownian motion parameters are reduced, the temperature distribution reduces.

Svante Arrhenius, a Swedish physicist, used the phrase energy for the first time in 1889. Activation energy is measured in KJ/mol and denoted by *Ea*, which means the minimum energy achieved by molecules/atoms to initiate the chemical process. For various chemical processes, the amount of energy activation is varying, even sometimes zero. The activation energy in heat transfer and mass transfer has its usages in chemical engineering, emulsions of different suspensions, food processing, geothermal reservoirs, etc. Bestman [25] published the first paper on activation energy with a binary chemical process. Discussion on the inclusion of chemical reaction into nanofluids flow and Arrhenius activation energy was determined by Khan et al. [26]. Zeeshan et al. [27] studied the Couette-Poiseuille flow with activation energy and analyzed convective boundary conditions. Bhatti and Michaelides [28] discussed the influence of activation energy on a Riga plate with gyrotactic microorganisms. Khan et al. [29] reveal that the impact of activation energy on the flow of nanofluid against stagnation point flow by considering it nonlinear with activation energy. Their investigation revealed that activation energy decline for the mass transfer phenomena. Hamid et al. [30] inquired about the effects of activation energy inflow of Williamson nanofluid with the influence of chemical reactions. The study concluded that the heat transfer rate in cylindrical surfaces declines when increasing the reaction rate parameter. Azam et al. [31] inquired about the impact of activation energy in the axisymmetric nanofluid flow. Waqas et al. [32] inquired the flow of Oldroyd-B bioconvection nanofluid numerically with nonlinear radiation through a rotating disc with activation energy.

Bioconvection characterizes the hydrodynamic instabilities and the forms of suspended biased swimming microorganisms. The hydrodynamics instabilities occur due to the coupling between the cell's swimming performance and physical features of the cell, i.e., fluid flows and density. For example, a combination of gravitational and viscous torques tend to swim the cells in the direction of down welling fluid. A gyrotactic instability ensues if the fluid is less dense than the cells. Bioconvection portrays a classical structure where a macroscopic mechanism occurs due to the microscopic cellular ensuing in relatively dilute structures. There is also the ecological impact for bioconvection and its mechanisms, which is promising for industrial development. In the recent era, many scientists have discussed the mechanism of bioconvection using nanofluid models. For instance, Makinde et al. [33] examined the nanofluid flow due to rotating disk and thermal radiation with titanium and aluminum nanoparticles. They showed that the base liquid thermal efficiency is remarkable when the nanoparticles of titanium alloy are introduced in contrast to the nanoparticles of aluminum alloy. Reddy et al. [34] studied the Maxwell thermally radiative nanofluid flow on a double rotating disk. Waqas et al. [35] examined the effect of thermally bioconvection Sutterby nanofluid flow between two rotating disks along with microorganisms. The fluid speed with mixed convection parameters grew quicker but delayed the magnetic field parameter and the Rayleigh number bioconvection. Some important studies on the bioconvection mechanism can be found from the list of references [36–39].

For many industrial applications such as the production of glass, furnaces, space technologies, comic aircraft, space vehicles, propulsion systems, plasma physics, and reentry aerodynamics in the field of aero-structure flows, combustion processes, and other spacecraft applications, the role of thermal radiation is significant. Raju et al. [40] examined the flow of convective magnesium oxide nanoparticles with nonlinear thermal convective over a rotating disk. Sheikholeslami et al. [41] presented the analysis of thermally radiative MHD nanofluid through the porous cavity. Muhammad et al. [42] analyzed the characteristics of thermal radiation for Powell-Eyring nanofluid flow with additional effects of activation energy. Aziz et al. [43] numerically analyzed hybrid nanofluid with entropy analysis, thermal radiation, and viscous dissipation. Mahanthesh et al. [44] investigated the significance of radiation effects of the two-phase flow of nanoparticles over a vertical plate. Jawad et al. [45] investigated the bio-convection nanofluid flow of Darcy law through a channel (Horizontal) with magnetic field effects and thermal radiation. Majeed et al. [46] thermally analyzed magnetized bioconvection flow with additional effects of activation energy. Numerous fresh developments on this topic can be envisaged through [47–52].

After studying the preexistent literature, it is noticed that there is no addition to the research of Reiner-Rivlin fluid flow between rotating circular plates filled with microorganisms and nanoparticles. In the present study, we assume that the flow in the tangential and axial direction. The Reiner-Rivlin nanofluid with motile gyrotactic microorganisms is filled between the pair of rotating plates. The thermally radiative Reiner-Rivlin fluid is electrically conducted under the existence of activation energy. The famous Differential Transform scheme is used to obtain the solution of the ordinary differential equations. Padé approximation is also applied to enhance the convergence rate of the solution obtained by the Differential Transform Method. The impact of various parameters in nanoparticle concentration, velocity, temperature, and motile microorganism function is analyzed thoroughly using graphs and tabular forms.

### **2. Physical and Mathematical Structure of Three-Dimensional Flow**

Let us anticipate incompressible three-dimensional, unsteady, axisymmetric squeezed film flow of Reiner-Rivlin nanofluid between a circular rotating parallel plate. The height of both plates is taken as - Γ (*t*) = *<sup>D</sup>*(−*β<sup>t</sup>* + 1)1/2 at time *t*. Let (*r*, *θ*, *z*) be the cylindrical polar coordinates with velocity field *V* = [*vr*, *vθ*, *vz*]. The lower circular plate is fixed while the upper circular plate is considered as moving towards the lower plate. The moving plate velocity is represented by - Γ (*t*). Both plates are rotating at a symmetric axis, which is characterized by *Z*-axis. The components of the magnetic field applied **H** on the moving plate in axial and azimuthal direction are:

$$
\widehat{H}\_{\theta} = \frac{rN\_0}{\mu\_2} \sqrt{\frac{D}{\widehat{\Gamma}\left(t\right)}},\\\widehat{H}\_z = -\frac{\beta M\_0 D}{\mu\_1 \widehat{\Gamma}\left(t\right)},\tag{1}
$$

Here *N*0, *M*0 in Equation (1) denotes the dimensionless quantities, which results - *Hθ*, - *Hz* in dimensionless, and the magnetized permeability of medium inside and outside of both plates are characterized by *μ*2 and *μ*1, respectively. In the case of liquid metals, *μ*2 = *μ* where *μ* indicates the free space permeability. *Hθ*, *Hz* on a fixed plate is expected to be zero. The extrinsic applied magnetic field **H** tends to generate an induced magnetic field - **<sup>B</sup>**(*<sup>r</sup>*, *θ*, *z*) having components - *Br*, - *Bθ*, - *Bz* between the two plates (see Figure 1). The temperature and the concentration at the lower plate is denoted as (*<sup>T</sup>*0, *<sup>C</sup>*0) while at the upper plate is taken as (*<sup>T</sup>*1, *<sup>C</sup>*1).

**Figure 1.** A physical structure for nanofluid flow between parallel circular plates in the existence of motile gyrotactic microorganisms and induced MHD.

### *2.1. Mathematical Modeling of Reiner-Rivlin Fluid*

The constitutive equation of Reiner-Rivlin fluid flow is defined as [53]:

$$
\pi\_{i\dot{j}} = -p\delta\_{i\dot{j}} + \mu\varepsilon\_{i\dot{j}} + \mu\_c\varepsilon\_{ik}\varepsilon\_{k\dot{j}\prime}\ \varepsilon\_{j\dot{j}} = 0,\tag{2}
$$

where *τij* represents stress tensor, *p* denotes pressure, *μ* denotes the viscosity coefficient, *μc* denotes cross-viscosity coefficient, *δij* denotes Kronecker symbol, and deformation rate tensor is represented by *eij* = *<sup>∂</sup>ui*/*∂xj* + *<sup>∂</sup>uj*/*∂xi* . Components of deformation rate tensor are:

$$\begin{aligned} \varepsilon\_{rr} = 2D\_2 \upsilon\_{r\prime} \ \varepsilon\_{\theta\theta} = 2\frac{v\_r}{r} \ \varepsilon\_{zz} = 2D\_4 \upsilon\_{z\prime} \ \varepsilon\_{r\theta} = \varepsilon\_{\theta r} = rD\_2 \left(\frac{v\_\theta}{r}\right) = D\_2 \upsilon\_\theta - \frac{v\_\theta}{r},\\ \varepsilon\_{z\theta} = \varepsilon\_{\theta z} = D\_4 \upsilon\_\theta \ \varepsilon\_{rz} = \varepsilon\_{zr} = D\_4 \upsilon\_r + D\_2 \upsilon\_z \end{aligned} \tag{3}$$

with the help of Equation (2), components of stress tensor are attained as

$$
\pi\_{\mathcal{I}\mathcal{I}} = -p + \mu \mathfrak{e}\_{\mathcal{I}\mathcal{I}} + \mu\_{\mathcal{C}} \left( \mathfrak{e}\_{\mathcal{I}\mathcal{I}}{}^2 + \mathfrak{e}\_{r\mathcal{I}}{}^2 + \mathfrak{e}\_{\mathcal{I}z}{}^2 \right), \tag{4}
$$

$$\tau\_{rr} = -p + 2\mu D\_2 v\_r + \mu\_c \left[ 4(D\_2 v\_r)^2 + \left( D\_2 v\_\theta - \frac{v\_\theta}{r} \right)^2 + \left( D\_4 v\_r + D\_2 v\_z \right)^2 \right],\tag{5}$$

$$
\pi\_{r\theta} = \pi\_{\theta r} = 0 + \mu \varepsilon\_{r\theta} + \mu\_{\varepsilon} (\varepsilon\_{r\tau} \varepsilon\_{r\theta} + \varepsilon\_{r\theta} \varepsilon\_{\theta\theta} + \varepsilon\_{r\overline{z}} \varepsilon\_{z\theta}),
\tag{6}
$$

$$\begin{array}{ll} \pi\_{r\theta} = \mu \left( D\_{2} \upsilon\_{\theta} - \frac{\upsilon\_{\theta}}{r} \right) + \\ & + \left( D\_{2} \upsilon\_{\theta} - \frac{\upsilon\_{\theta}}{r} \right) \left( 2 \frac{\upsilon\_{r}}{r} \right) + (D\_{4} \upsilon\_{\theta}) \left( D\_{1} \upsilon\_{r} + D\_{2} \upsilon\_{z} \right) \end{array} \tag{7}$$

$$
\pi\_{rz} = \mu e\_{rz} + \mu\_c (e\_{rr} e\_{r\overline{z}} + e\_{r\theta} e\_{\theta\overline{z}} + e\_{r\overline{z}} e\_{\overline{z}\overline{z}}),
\tag{8}
$$

$$\begin{aligned} \pi\_{rz} &= \mu (D\_4 v\_r + D\_2 v\_z) + \mu \frac{\mu\_c [2 (D\_2 v\_r)(D\_4 v\_r + D\_2 v\_z) \\ &+ (D\_2 v\_\theta - \frac{v\_\theta}{r})(D\_4 v\_\theta) + 2 (D\_4 v\_z)(D\_4 v\_r + D\_2 v\_z)]}{(\frac{D\_4}{r})} \end{aligned} \tag{9}$$

$$
\pi\_{\theta\theta} = -p + \mu \varepsilon\_{\theta\theta} + \mu\_{\mathfrak{C}} \left( \left. \varepsilon\_{r\theta} \right|^2 + \left. \varepsilon\_{\theta\theta} \right|^2 + \left. \varepsilon\_{z\theta} \right|^2 \right), \tag{10}
$$

$$\pi\_{\theta\theta} = -p + \mu \left( 2\frac{v\_r}{r} \right) + \mu\_c \left[ \left( D\_2 v\_\theta - \frac{v\_\theta}{r} \right)^2 + 4 \left( \frac{v\_r}{r} \right)^2 + \left( D\_4 v\_\theta \right)^2 \right],\tag{11}$$

$$
\pi\_{\theta z} = \mu c\_{\theta z} + \mu\_{\varepsilon} (\varepsilon\_{\theta r} c\_{rz} + \varepsilon\_{\theta \theta} c\_{\theta z} + \varepsilon\_{\theta z} c\_{zz}),
\tag{12}
$$

$$\tau\_{\theta z} = \tau\_{z\theta} = \mu (D\_4 v\_{\theta}) + \mu\_\varepsilon \left[ \left( D\_2 v\_{\theta} - \frac{v\_{\theta}}{r} \right) (D\_4 v\_r + D\_2 v\_z) + 2 (D\_4 v\_{\theta}) \left( \frac{v\_r}{r} + D\_4 v\_z \right) \right], \tag{13}$$

where *D*1 = *∂∂t* , *D*2 = *∂∂r*, *D*3 = *∂∂θ* , *D*4 = *∂∂z* .
